Print browse tree in dump by default

The tree isn't that large since only the current path is cached.
This will help with debugging browse tree related issues where
currently all we can do is read snoop logs (if enabled) and trust
the UI.

Tag: #refactor
Bug: 292261138
Test: atest BluetoothInstrumentationTests
Change-Id: I38dc48893927b832f741d40b6c2a2593a31c9ae4